1. ISSPC 2011 - Rio de Janeiro1 T2 Control Charts with Variable Dimension Eugenio K. Epprecht - PUC-Rio, Brazil Francisco Aparisi – UPV, Spain Omar Ruiz - ESP del Litoral, Equador

2. ISSPC 2011 - Rio de Janeiro2 Outline Introduction –Basic idea –Related works Standard T 2 control chart procedure Charts proposed –Variable-Dimension T 2 control chart (VDT2) –Double-Dimension T 2 control chart (DDT2) ARL computation –ARL of the VDT2 chart –ARL of the DDT2 control chart Optimization and Performance Comparison –User Interface –Results –Performance Comparison Another version of the problem (and software) Continuation of the research References

3. ISSPC 2011 - Rio de Janeiro3 Introduction - basic idea T 2 control chart number of variables to monitor: variable p = p1 + p2 p1 out of the p variables : easy or cheap to monitor/measure remaining p2 variables : difficult/expensive to monitor/measure  Idea: controlling sometimes only the p1 (“cheap”) variables, and only when the process seems to have a problem: controlling the full set of p variables.

4. ISSPC 2011 - Rio de Janeiro4 Examples Expensive electronic component with 3 (correlated) quality variables to be monitored. –X1 and X2: voltages, cheap and easy to measure –X3: measurement is destructive (e.g. the voltage that will burn a part of component) Other cases: –some variables could need, for instance, a laboratory analysis (difficult, slow, expensive, etc.)

5. ISSPC 2011 - Rio de Janeiro5  Similar to adaptive control charts, but instead of varying sample size, sampling interval and/or control limits: varying the number of variables to consider

6. ISSPC 2011 - Rio de Janeiro6 Related works Univariate: –Surrogate monitoring (Costa and De Magalhães, 2005) –Double-stage sampling (by attribute – by variables) (Costa et al., 2005) Multivariate: –T 2 – reduces the number of charts –PC’s – reduce the number of dimensions of the space...but one still measures all p variables! –González and Sanchez (2010) – reduce the number of variables (but always! and according to other criterion)

7. ISSPC 2011 - Rio de Janeiro7 Our goal: to reduce the number of variables to measure (on average)  reducing costs (and/or time)

8. ISSPC 2011 - Rio de Janeiro8 Standard T 2 control chart procedure samples of size n, of the p-dimensional vector (X1, X2,... Xp) where: : in-control mean vector : in-control covariance matrix

9. ISSPC 2011 - Rio de Janeiro9 Standard T 2 control chart procedure – (cont.) When the process is in control ( )  When the process is out of control ( )  where or d : Mahalanobis’ distance of with respect to

10. ISSPC 2011 - Rio de Janeiro10 Charts proposed VDT2 control chart (variable dimension) DDT2 control chart (double dimension)

11. ISSPC 2011 - Rio de Janeiro11 Variable-Dimension T 2 control chart (VDT2) We may have two warning limits (w1 and w), but our results show that although a chart with two warning limits shows better performance, the improvement is not large w w CLp1 CLp p1 p2

12. ISSPC 2011 - Rio de Janeiro12 Double-Dimension T 2 control chart (DDT2) if T 2 p1 ≤ w  process considered in control if T 2 p1 ≥ CL p1  process declared out of control if w ≤ T 2 p1 ≤ CL p1  remaining p2 variables are measured and combined with the measurements of the set of p1 variables  T 2 p T 2 p plotted in the vertical of T 2 p1 and compared with CL p  decision to measure all variables: taken for the same sample, not waiting until the next sampling time

13. ISSPC 2011 - Rio de Janeiro13 Double-Dimension T 2 control chart (DDT2) w w CLp1 CLp p1 p1+p2 p1 p1+p2

14. ISSPC 2011 - Rio de Janeiro14 ARL computation ARL of the VDT2 chart: Markov’s chain model -- similar to ARL of adaptive (variable) Shewhart charts (VSS, VSI, VSSI, Vp) (Reynolds et al., 1988; Aparisi, 1996; Costa, 1999; Epprecht et al., 2003) In the case of only one warning limit: State 1: (and the next sample will contain only p1 variables) State 2: (and the next sample will contain all p variables) State 3: (signal; absorbing state)

15. ISSPC 2011 - Rio de Janeiro15 ARL of the VDT2 chart (cont.) Transition probability matrix for shift d : For example: where

16. ISSPC 2011 - Rio de Janeiro16 ARL of the VDT2 chart (cont.) Zero-state ARL: where, I : 2  2 identity matrix, : vector of initial state probabilities, : out-of-control probability transition matrix between transient states

17. ISSPC 2011 - Rio de Janeiro17 ARL of the VDT2 chart (cont.) Zero-state ARL: where, I : 2  2 identity matrix, : vector of initial state probabilities, : out-of-control probability transition matrix between transient states For the ARL 0, use Q 0 instead of Q d in the formula above

18. ISSPC 2011 - Rio de Janeiro18 ARL of the VDT2 chart (cont.) Steady-state ARL: : in-control probability transition matrix between transient states

19. ISSPC 2011 - Rio de Janeiro19 ARL of the DDT2 control chart Calculations analogous to those of the ARL of –double-sampling control charts (Daudin, 1992; Champ& Aparisi, 2008; De Araujo Rodrigues et al., 2011) –double-sampling procedures for acceptance sampling ARL = 1 / (1 – P(no signal))

20. ISSPC 2011 - Rio de Janeiro20 ARL of the DDT2 control chart Calculations analogous to those of the ARL of –double-sampling control charts (Daudin, 1992; Champ& Aparisi, 2008; De Araujo Rodrigues et al., 2011) –double-sampling procedures for acceptance sampling ARL = 1 / (1 – P(no signal)) ARL results for the DDT2 chart are underway

21. ISSPC 2011 - Rio de Janeiro21 Optimization and Performance Comparison Software for design optimization — using Genetic Algorithms Execution time < 5 min Case: –p1 = 2, p2 = 2, p = 2 + 2 = 4. –Desired in-control ARL: 400 –Shifts specified: dp1 = 0.5 (when p1 = 2 variables are measured) dp = 1 (when all p = 4 variables are monitored)

22. ISSPC 2011 - Rio de Janeiro22 User interface (for VDT2 chart with one warning limit)

23. ISSPC 2011 - Rio de Janeiro23 Results w = 2.61, (which has an in-control probability “left tail” equal to 0.729) CLp1 = 40.95, with right tail probability = 0  no control limit is needed for p1 variables  common result for this scheme, which simplifies the use of the chart, because only the warning limit and one control limit (for the p variables) are required. CLp = 14.44 with a right tail probability of 0.00602. In-control ARL = 399.94. ARL for the given shift is 101.31

24. ISSPC 2011 - Rio de Janeiro24 Comparing this performance with the performances of: T2 control chart where always p1 variables are measured T2 control chart where always all the p variables are monitored (with ARL0 = 400) T2 (p1 variables): out-of-control ARL = 216.90 T2 (p variables): out-of-control ARL = 107.86 (VDT2: ARL = 101.31)  ARL improvement is marginal

25. ISSPC 2011 - Rio de Janeiro25 However: sampling cost of VDT2 << sampling cost of T2p % of times than we must sample all the variables when the process is in control : 42% Average cost of sampling of the VDT2: -cost of measuring the p 1 variables: c 1 -cost of measuring the p 2 variables: c 2  average cost per sample: c 1 + 0.42c 2 (against a fixed cost of c 1 + c 2 of the T 2 p)  relative economy of 58c 2 / (c 1 + c 2 )% per sample)

26. ISSPC 2011 - Rio de Janeiro26 Another version of the problem (and software) Average cost per sample: c 1 + 0.42c 2 or, equivalently, 0.58c 1 + 0.42(c 1 + c 2 ) Depending on the values of c 1 and c 2, this may be still high  in other version of the problem: % of times that all p variables are measured (when the process is in control): constraint for the optimization

27. ISSPC 2011 - Rio de Janeiro27 Example If this percentage is fixed at 20%, the sftw returns: w = 3.82 CLp1 = 24.35 (with right tail probability = 0.00000470  practically the control limit can be set to infinite) CLp = 12.80, (right tail probability = 0.01229719) In-control ARL = 400. OOC ARL = 105.74  with only a marginal reduction in performance, the sampling cost could be significantly reduced

28. ISSPC 2011 - Rio de Janeiro28 Continuation Optimization considering costs (and/or time) to measure (under a given cost per time and a given ATS 0, minimize the AATS) Enhanced schemes

29. ISSPC 2011 - Rio de Janeiro29 REFERENCES Aparisi, F. (1996). Hotelling’s T2 control chart with adaptive sample sizes. IJPR, 34, 2853-2862. Aparisi, F., De Luna, M., Epprecht, E. (2010). Optimisation of a set of Xbar or principal components control charts using genetic algorithms. IJPR, 48(18), 5345-5361. Champ, C. W. and Aparisi, F. (2008). Double Sampling Hotelling's T2 Charts. QREI, 24, 153-166. Costa, A. F. B. (1999) Xbar charts with variable parameters, JQT, 31(4), p. 408-416 Costa, A.F.B. and De Magalhães, M. S. (2005). Economic design of two-stage Xbar charts: the Markov-chain approach. IJPE 95(1), p.920. Costa, A. F. B., De Magalhaes, M. S. and Epprecht, E. K. (2005). The Non-central Chi-square Chart with Double Sampling. Brazilian Journal of Operations and Production Management 2(1), p. 57-80. Daudin, J. J. (1992) Double sampling Xbar charts, JQT 24(2), pp. 78-87 Epprecht, E. K.; Costa, A.F.B.; Mendes, F.C.T. (2003) Adaptive Control Charts for Attributes. IIE Transactions, 35 (6), p. 567-582. De Araujo Rodrigues, A.; Epprecht, E.K. ; De Magalhães, M.S. (2011) Double-sampling control charts for attributes. JAS, 38, p. 87-112. González, I. and Sanchez, I. (2010). Variable Selection for Multivariate Statistical Process Control. JQT 42(3), p. 242-259. Reynolds, M. R. Jr., Amin, R. W., Arnold, J. C. & Nachlas, J. A. (1988). Xbar charts with variable sampling intervals, Technometrics 30(2), p.181-192.